傅里叶光学C2AnalysisofTwo-DimensionalSignalsandSystems.pptVIP

CHAPTER 2 Analysis of Two-Dimensional Signals and Systems ?Some Frequently Used Functions ?Convolution ?Fourier Transform ?Some Useful Fourier Transform Pairs Frequently Used Functions -Rectangle function Sinc function Signum function Triangle function Circle function Dirac delta function Comb function Convolution The convolution of f1(x,y) and f2(x,y) is defined as Autocorrelation The autocorrelation of X(x,y) is defined as Fourier Series Representation of Periodic Signals Let x(t) be a periodic signal with period T, i.e., The Fourier Series Then, x(t) can be expressed as Dirichlet Conditions Fourier Transform Periodic signals can be represented with the Fourier series Aperiodic signals can be analyzed in terms of frequency components also The major difference from the line spectra of periodic signals is that the spectra of aperiodic signals are defined for all real values of the frequency variable not just for a discrete set of values One-dimensional Fourier transform Let period T??, periodic function becomes aperiodic function Fourier analysis in two dimensions Fourier analysis is a mathematical tool of great utility in the analysis of both linear and nonlinear phenomena It is widely used in the study of electrical networks and communication systems In optics, it appears in form of two dimensions Definition of Fourier Transform Fourier transform of function g(x,y) is defined by Transform pairs for some functions Existence Conditions of Fourier Transform Original function g(x,y) must be absolutely integrable over the infinite (x,y) plane g(x,y) must have only a finite number of discontinuities and a finite number of maxima and minima in any finite rectangle g(x,y) must have no infinite discontinuities Comparison between existence conditions Absolutely integrable, finite number of maxima and minima, finite number of discontinuities, no infinite discontinuities Fourier series needs above conditions over one period Fourier transform needs above conditions